Non-constant discounting and consumption, portfolio and life insurance rules

Highlights

• Consumption, portfolio and life insurance rules under hyperbolic preferences.

• Results are obtained and compared for three different behavior strategies.

• Qualitative behavior of the consumption is maintained with hyperbolic preferences.

Abstract

Consumption, portfolio and life insurance rules are studied for an investor with an arbitrary but known distribution of lifetime with time-inconsistent preferences. Solutions are found for naive and sophisticated agents for the family of CARA and CRRA utility functions. Effects of non-constant discounting are illustrated numerically.

Introduction

Intertemporal decision problems have been revised in the recent literature since the classical exponential discounting has been found to be unrealistic, at least in some situations, as pointed out in the comprehensive work of Frederick et al. (2002). Hyperbolic discounting has been introduced in order to account for situations where the time preferences of the decision maker change depending on whether these decisions are short-term or long-run actions. In a continuous time setting, several recent papers (e.g., Karp, 2007) have introduced variable rates of time preference. As preferences change continuously over the time horizon, it is usual to refer to the $t$-agent as the decision maker that makes a decision at time $t$ in order to differentiate her from the $t^{\prime }$-agent, i.e., the same agent making a decision at a different moment $t^{\prime }$ (Strotz, 1956). If the agent is able to precommit her future behavior we obtain the so-called precommitment solution. On the contrary, under the assumption of no precommitment, agents can act in a naive or a sophisticated way. Naive agents make decisions without taking into account that their preferences will change in the future and, consequently, they will be continuously modifying their choices for the future. On the contrary, a sophisticated agent, in the search for a time-consistent policy, will take into account that her future preferences will be different, and she will constrain her current decisions to her future behavior. We refer to Yong (2012) and references therein for a study of the corresponding equilibrium conditions in a stochastic setting.

In this paper we depart from the work of Pliska and Ye (2007) where optimal consumption, portfolio and life insurance rules were studied for an investor with an arbitrary but known distribution of lifetime, and generalize the investor’s time preferences by considering a non-constant discount¹ rate within a stochastic framework, extending previous results from Marín-Solano and Navas (2010). Within this setting, we study precommitment, naive and sophisticated solutions for CARA and CRRA utility functions. For some discount functions, we characterize sophisticated solutions as the solution to a system of two ordinary differential equations that can be solved numerically. As in the standard discounting model, when only hyperbolic discounting is considered, our numerical results confirm the qualitative behavior found in the standard discounting model, in contrast to other results in the literature (see discussion in Section 3).